Mathematical Mysteries: Trisecting the Angle

Bisecting a given angle using only a pair of compasses and a straight edge is easy. But trisecting it - dividing it into three equal angles - is in most cases impossible. Why?

Bisecting an angle

If we have a pair of lines meeting at a point O, and we want to bisect the angle between them, here's how we do it.

Bisecting angle AOB using straight edge and compasses.

Put the point of the compass at O and draw a circle (of any radius you like). This is the blue arc in the diagram. The circle will cross the two lines at two points: call these A and B.

Now put the point of the compass at A and draw an arc of a circle, as shown in the diagram. Without changing the radius at which the compass is set, move its point to B and draw another arc of a circle. These are the red arcs in the diagram. The point where the two arcs cross, P, can then be joined to O using the straight edge (the green line), and angle POB is half of angle AOB. If the
arcs don't cross, then of course you need to use larger circles!

Can you prove that this procedure works, using similar triangles? The ancient Greeks certainly knew how to do this.

Trisecting an angle

What about trisecting an angle? Why is it so difficult? There are a few special cases of angles for which it can be done - for example, pi/2 radians (90 degrees). For the general case, however, the Greeks couldn't work out how to do it, despite expending vast amounts of energy on the problem.

Trisecting an arbitrary angle can be done if you cheat by using a measuring ruler instead of a plain straight edge (you can find out how in the sci.math FAQ file), or even if you simply draw two little marks on your straight edge. However, to "play by the rules" you aren't allowed any marks on the straight edge at all - it must be
completely blank.

The problem of whether trisection could be done in the general case remained a mathematical mystery for millennia - it was only in 1837 that it was eventually proved to be impossible by Pierre Wantzel, a French mathematician and expert on arithmetic. This was a great achievement for a man of 23, who
subsequently died at the tragically young age of 33.

So why is it impossible? Pierre showed that the two problems of trisecting an angle and of solving a cubic equation are equivalent. Moreover, he showed that only a very few cubic equations can be solved using the straight-edge-and-compass method - most cannot. He thus deduced that most angles cannot be trisected.

Footnote : The real mystery

Despite the fact that Wantzel's proof means that we now know that it's impossible to trisect a general angle, people keep trying. The mathematics department where PASS Maths is based receives quite serious letters every so often from individuals who think they've cracked the problem, offering famous people in the department the opportunity to "buy" the "proof" (sometimes for large
amounts of money!). One such letter even missed out several pages of the "proof" on the grounds of security of the writer's copyright!

Needless to say, all these so-called proofs contain flaws and are worthless. If the writers really wanted to convince anyone that it is possible to trisect the angle, their time would be better spent trying to find an error in Wantzel's proof. The real mystery here is why people keep trying to solve the problem in the face of a proof of its impossibility. How would you reply to
them?

Comments

1) Draw AB
2) Draw AC orbitrarily
3) With any radius with compass mark D along AC and E along AB
4) Join DE
Problem :
We have to divide DE to 3 equal parts
1) With D as center mark DF (any distance) arc and with same radius and compass mark FG with F as center and
GH with G as center along AC
2) Join HE
3) With GK as radius mark a circle with E as center . join GL (L is the outer most point in circle)
4) with the top most point L joint G & L with a line
5) Similarly with FN as radius draw a circle with center as I and join MF intersecting DE at J (J is the outer most point in circle)
6) Join AJ, AI
7) Hence angle BAC is divided into 3 parts

WITHOUT GIVING ANY PREVIOUS HINTS, YOU HAVE USED POINTS LIKE 'K', 'N', 'I' AND 'M' WHILE SAYING GK, FN AND MF. SIMILARLY WHILE SAYING 'OUTER MOST POINT OF A CIRCLE' YOU SHOULD HAVE BEEN MORE PRECISE AS THERE ARE TWO SUCH POINTS FOR A POINT LYING OUTSIDE THE CIRCLE. I THINK THERE ARE EASIER WAYS OF DIVIDING A LINE INTO 3 EQUAL PARTS.
BUT MOST IMPORTANT: DID YOU MEASURE THE THREE ANGLES CREATED AT 'A'? CAN YOU PROVE THAT THE ANGLES ARE OF EQUAL MEASURE?

In regards to people tackling a proof trisecting an angle, a case can be made that it isn’t always a worthless endeavor. In my case I am looking at it as a way to keep my mind and thinking active.
The easiest thing to do is to prove that it is impossible to trisect any angle precisely. However there is still the question of is there any method to effectively trisect any angle to an acceptable accuracy in a limited number of steps using only a compass and unmarked straight edge. This is a different issue and is a valid question.

Actually the aphorism is that it's not possible to prove a negative (and even that is sometimes wrong). Proving that things are impossible is done all the time... it is the basis, for example, of reductio absurdum or proof by contradiction.